Theorem llynlly 23460

Description: A locally 𝐴 space is n-locally 𝐴: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
llynlly	⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ 𝑛-Locally 𝐴)

Proof of Theorem llynlly

Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		llytop 23455	. 2 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
2		llyi 23457	. . . . 5 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
3		simpl1 1198	. . . . . . . 8 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
4	3, 1	syl 17	. . . . . . 7 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
5		simprl 776	. . . . . . 7 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝐽)
6		simprr2 1229	. . . . . . 7 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑢)
7		opnneip 23102	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽 ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
8	4, 5, 6, 7	syl3anc 1379	. . . . . 6 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
9		simprr1 1228	. . . . . . 7 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ⊆ 𝑥)
10		velpw 4534	. . . . . . 7 ⊢ (𝑢 ∈ 𝒫 𝑥 ↔ 𝑢 ⊆ 𝑥)
11	9, 10	sylibr 235	. . . . . 6 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝒫 𝑥)
12	8, 11	elind 4129	. . . . 5 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
13		simprr3 1230	. . . . 5 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝐽 ↾t 𝑢) ∈ 𝐴)
14	2, 12, 13	reximssdv 3157	. . . 4 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)
15	14	3expb 1126	. . 3 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)
16	15	ralrimivva 3182	. 2 ⊢ (𝐽 ∈ Locally 𝐴 → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)
17		isnlly 23452	. 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))
18	1, 16, 17	sylanbrc 589	1 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ 𝑛-Locally 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4529  {csn 4555  ‘cfv 6485  (class class class)co 7356   ↾_t crest 17374  Topctop 22876  neicnei 23080  Locally clly 23447  𝑛-Locally cnlly 23448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-top 22877  df-nei 23081  df-lly 23449  df-nlly 23450

This theorem is referenced by:  llyssnlly  23461  efmndtmd  24084